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A Variational Loop Shrinking Analogy for Handle and Tunnel Detection and Reeb Graph Construction on Surfaces

MPS-Authors
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Seidel,  Hans-Peter
Computer Graphics, MPI for Informatics, Max Planck Society;

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Zayer,  Rhaleb
Computer Graphics, MPI for Informatics, Max Planck Society;

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arXiv:2105.13168.pdf
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Citation

Weinrauch, A., Seidel, H.-P., Mlakar, D., Steinberger, M., & Zayer, R. (2021). A Variational Loop Shrinking Analogy for Handle and Tunnel Detection and Reeb Graph Construction on Surfaces. Retrieved from https://arxiv.org/abs/2105.13168.


Cite as: https://hdl.handle.net/21.11116/0000-0009-70EE-D
Abstract
The humble loop shrinking property played a central role in the inception of
modern topology but it has been eclipsed by more abstract algebraic formalism.
This is particularly true in the context of detecting relevant non-contractible
loops on surfaces where elaborate homological and/or graph theoretical
constructs are favored in algorithmic solutions. In this work, we devise a
variational analogy to the loop shrinking property and show that it yields a
simple, intuitive, yet powerful solution allowing a streamlined treatment of
the problem of handle and tunnel loop detection. Our formalization tracks the
evolution of a diffusion front randomly initiated on a single location on the
surface. Capitalizing on a diffuse interface representation combined with a set
of rules for concurrent front interactions, we develop a dynamic data structure
for tracking the evolution on the surface encoded as a sparse matrix which
serves for performing both diffusion numerics and loop detection and acts as
the workhorse of our fully parallel implementation. The substantiated results
suggest our approach outperforms state of the art and robustly copes with
highly detailed geometric models. As a byproduct, our approach can be used to
construct Reeb graphs by diffusion thus avoiding commonly encountered issues
when using Morse functions.